AI 中文总结
研究在特定域F上2n维向量空间V中，辛形式仿射空间在元素个数多于2n - 2且特征不为2时，具有临界维数n(n - 1)的空间分类，归结为F上非迷向二次型及F二次扩张上非迷向埃尔米特型分类，二次闭域同余意义下恰有一解。
AI 中文摘要
设F为一个域，V是F上的2n维向量空间。在之前的一篇文章中，我们证明了如果F的元素个数多于2n - 2，那么V上辛形式的仿射空间的最大可能维数是n(n - 1)。在此，在相同的基数假设下，我们研究具有临界维数n(n - 1)的空间。特别地，如果F的特征不为2，这些空间在同余意义下的分类可归结为：(1) F上非迷向二次型在等价和乘以非零标量意义下的分类；(2) F的所有二次扩张上非迷向埃尔米特型在等价和乘以 - 1意义下的分类。特别地，对于二次闭域，证明了在同余意义下恰好有一个解。
英文摘要
Let F be a field and V be a 2n-dimensional vector space over F. In a previous article, we have proved that if F has more than 2n-2 elements then the greatest possible dimension for an affine space of symplectic forms on V is n(n-1). Here, under the same cardinality assumption we study the spaces that have the critical dimension n(n-1). In particular, if the characteristic of F is not 2 the classification of these spaces up to congruence is reduced to: (1) the classification of nonisotropic quadratic forms over F, up to equivalence and multiplication with a nonzero scalar; (2) the classification of nonisotropic Hermitian forms over all quadratic extensions of F, up to equivalence and multiplication by $-1$. In particular, for quadratically closed fields it is shown that there is exactly one solution up to congruence.
Comments89 pages (including a table of contents)
